H pylori: could it cause black phlegm, shortness of breathe, and chest pain?

Possibly. If you have an H pylori ulcer with reflux this could cause your symptoms. Please see your doctor to check for this as well as look for other causes. Good luck.
No. H Pylori infection affect the stomach, causing gastritis, stomach ulcers, duodenal ulcers, if left untreated H.Pylori has been associated with a type of lymphoma of the stomach and stomach cancer. H. Pylori does not cause black phlegms, shortness of breath or chest pain. Contact your family doctor ASAP for that.